Biomedical Engineering Reference
In-Depth Information
Similarly to the power spectrum, cross-spectrum is usually computed by means of
the Fourier transforms
X
and
Y
of the signals
x
and
y
:
1
T
X
Y
∗
(
S
xy
(
f
)=
lim
T
(
f
,
T
)
f
,
T
)
(3.5)
→
∞
The fact that we have only a limited data window of length
T
has the same con-
sequences as in cases of power spectra estimation (see Sect. 2.3.2.1). Usually non-
rectangular window functions are used and smoothing is applied. Computation of
cross-spectrum is implemeted in MATLAB Signal Processing Toolbox as
cpsd
func-
tion. It estimates the cross power spectral density of the discrete-time signals using
Welch's averaged, modified periodogram method of spectral estimation.
S
xy
(
f
)
is a complex value consisting of real and imaginary parts:
(
)=
(
)(
)+
(
)(
)
S
xy
f
Re
S
xy
f
iIm
S
xy
f
(3.6)
In polar coordinates it can be expressed by the formula:
e
i
Φ
xy
(
f
)
,
S
xy
(
f
)=
|
S
xy
(
f
)
|
(3.7)
)
|
=
Re
2
2
where
|
S
xy
(
f
(
S
xy
)
(
f
)+
Im
(
S
xy
)
(
f
)
is
a
modulus
and Φ
xy
(
f
)=
tan
−
1
Im
(
S
xy
)
Re
is a phase.
Coherence is a measure which is often used in biomedical application. It is ex-
pressed by the formula:
(
S
xy
)
S
xy
(
f
)
γ
xy
(
f
)=
S
x
(3.8)
(
f
)
S
y
(
f
)
)
≤
S
x
where
S
x
and
S
y
are spectra of signals
x
and
y
.Since
S
xy
(
f
(
f
)
S
y
(
f
)
, func-
tion γ
xy
1. Coherence shows the relation between two signals in frequency and
phase. The square of the coherence measures the spectral power in a given frequency
common to both signals.
The above formula defines ordinary (bivariate) coherence. If a data set contains
more than two channels, the signals can be related with each other in different ways.
Namely, two (or more) signals may simultaneously have a common driving input
from the third channel. Depending on the character of relations between channels,
some of them may be connected directly with each other and some connections can
be indirect (through other channels). To distinguish between these situations partial
and multiple coherences were introduced.
The construction of partial coherence relies on subtracting influences from all
other processes under consideration. For three channels partial coherence is defined
as a normalized partial cross spectrum:
(
f
)
≤
|
S
xy
|
z
(
f
)
|
κ
xy
|
z
(
f
)=
(3.9)
(
S
xx
|
z
(
f
)
S
yy
|
z
(
f
)
where partial cross-spectrum is defined as:
S
−
1
zz
S
xy
|
z
(
f
)=
S
xy
(
f
)
−
S
xz
(
f
)
(
f
)
S
zy
(
f
)
(3.10)
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